This question is about physical theory, but my question is pure nathematical, so I post it here. I don't think you have to know physics in order to answer it.
I am studying liquid crystal theory with the book Kleman, Lavrentovich, Soft Matter Physics.
In the Ericksen-Leslie theory, Frank-OSeen energy is:
$$ f=0.5*(K_1*div^2 (n)+K_2 *(n*curl(n))^2+K_3*(n \times curl(n) )^2) $$
Here $\overrightarrow{n}=\overrightarrow{n}(x,y,z,t)$ is director (almost like a vector, for this problem you can treat it is a vector field), and summation over repeated indexes is used. $K_1,K_2,K_3$ are constants. "$*$" denotes scalar product (or product of two scalar quantities), "$ \times$" denotes vector product.
Later they take a full derivative with respect to time:
$$ {{df} \over {dt}}= {\partial f \over \partial n_i} {dn_i \over dt} + {\partial f \over \partial (\partial n_i / \partial x_j)} {d \over dt} {\partial n_i \over \partial x_j} $$
Summation over repeated indexes is used. Variables $x,y,z = x_1, x_2,x_3$ respectively are spatial variables, as usual in tensor algebra.
The above formula for $df/dt$ looks like the famous formula for the total derivative, but if the independent variables were $t$ and spatial derivatives of the director components.
1) Am I write that expression for $df/dt$ contains a sum of 10 terms?
2) How can I make sure (proove), that I indeed can use above 10 variables as independent, instead of standard 4 variables $t,x,y,z$.
Note: I think, the precise formula for $f(\overrightarrow{n})$ is not really important. I posted it just in case, so you can better understand what is going on.
Hope, that my question is clear. Thanks